Particles and Strings –
Probing the Structure of Matter and Space-Time
Jan Louis
University Hamburg
DPG-Jahrestagung, Berlin, March 2005
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Physics in the 20thcentury
Quantum Theory (QT) General Relativity (GR)
Planck, Bohr, Heisenberg, ... Einstein
Physics of small scales Physics of large scales
QT: asks about origin and structure of matter
GR: asks about structure of space-time and history of our universe
But: so far no unified theory valid at all length scales
– no ‘Theory of Everything’ – no ‘Quantum Gravity’
String Theory is possible candidate
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Quantum Theory → Particle Physics
➪ 20th century:
probing nature at increasingly smaller length scales
➪ led by two questions:
1. what are the constituents of matter?
2. which theory governs their interactions?
➪ answers changed with time/length scale
1. atoms → protons/neutrons/electrons → quarks & leptons → ?
2. quantum mechanics → quantum field theory → ?
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Particle Physics → Standard Model (SM)
Today both questions are answered by the SM
1. constituents of matter:
3 families of quarks & leptons (spin = 1/2)
2. interactions: governed by
– local quantum field theory
– symmetry principle: the non-Abelian gauge symmetry
G = SU(3) × SU(2) × U(1)
⇒ gauge bosons: gluons, W±, Z0, photon (spin = 1)
– G is spontaneously broken by Higgs Boson (spin = 0)
⇒ origin of mass
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Limits of the Standard Model
➪ experimental situation:
• SM has been fantastically confirmed
• no direct observation of Higgs boson yet ⇒ LHC
➪ questions unanswered:
• what determines G and the spectrum of particles ?
what determines masses and couplings of quarks & leptons ?
• gravitational interaction cannot be consistently turned on !
• origin of dark matter ?
➪ common belief:
SM is only an ‘effective theory’ above some scale l ≤ 10−18m
Below this scale: new phenomena and new theory.
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Generalization: Supersymmetry[Wess, Zumino]
enlarge the symmetry principle of the Standard Model:
{Q, Q†} = γµpµ
extension of the Poincare-algebra (Q is generator)
⇒ symmetry among fermions and bosons
⇒ doubling of the particle spectrum
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Properties:
➪ quantum corrections are “tamed”
⇒ light Higgs boson is ‘natural’ and predicted
⇒ strongly coupled QFT can be better controlled
➪ consistent with electro-weak precision data
➪ dark matter candidate: lightest supersymmetric particle (LSP)
➪ gauge coupling unification
⇒ Experimental verification: LHC, ILC
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Generalization: Grand Unified Theories[Georgi, Glashow]
further generalization of the symmetry principle of the SM:
GGUT ⊃ GSM = SU(3) × SU(2) × U(1)
(e. g.: GGUT = SU(5), SO(10), E6)
necessary condition: unification of the coupling constants
⇒ supersymmetric GUTs at scale MGUT ∼ 3 · 1016GeV
Properties:
• predicts decay of the proton
• suggests light neutrinos mν ∼M2
Z0
MGUT∼ O(10−3eV)
⇒ MGUT is new scale in nature
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General Relativity (GR)[Einstein]
➪ GR is a field theory of the gravitational interaction
Energy & Momentum
⇓
Curvature of space-time
⇓
Gravitational Interaction
Einstein equations: Rµν −1
2gµνR − Λgµν = GN Tµν
(Λ: cosmological constant, GN : Newton’s gravitational constant)
➪ GR successfully governs cosmology and astrophysics
⇒ Standard Model of Cosmology
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Limits of General Relativity
➪ experimental situation: GR has been fantastically confirmed
• correction to Newton law, deviation of light in grav. field
• existence of Black Holes,
• expanding universe, ...
➪ questions unanswered:
• what sets the value of GN – why is it so small?
what sets the value of Λ – why is it so small and positive ?
• physics of singularities (Big Bang and Black Holes) ?
quantum theory of gravity ?
➪ common belief:
GR is only an ‘effective’ theory valid on macroscopic scales
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Where is the Problem?
At what scale is a quantum gravity necessary?
λCompton =~
Mc≈ RSchwarz =
MG
c2
⇒Planck mass: MPL =
√
~cG
, EPL = c2MPL ≈ 1019GeV
Planck length: lPL =√
G~
c3 ≈ 10−35m
relevant in
• (very) early universe t = tPL = lPL/c ≈ 10−43s after big bang
• physics of black holes
belief:
at length scales l ≈ lPL =√
~Gc3 ≈ 10−35m
a completely new concept is necessary to describe nature.
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(Perturbative) String Theory
Idea: point-like objects → strings
Strings move in D-dimensional Minkowskian background.
(pert.) string theory is quantum theory of extended objects (strings).
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Proposal:
fundamental constituents of matter and space-time are strings
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Quantum excitations:
➪ finitely many massless excitations:
s = 2 → graviton ⇒ gravity cannot be turned off
s = 3/2 → gravitino ⇒ supersymmetric spectrum
s = 1 → gauge bosons
s = 1/2 → fermions (quarks & leptons)
s = 0 → Higgs, ...
➪ infinitely many massive excitations
M ∼ n · MS ,
MS = characteristic scale of string theory (tension of the string)
MS ∼ MPl (from coupling of graviton)
⇒ soft UV behavior
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Back to Strings: Interactions
∼ gs
gs = e−〈φ〉 is coupling constant, φ is scalar field (dilaton)
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scattering amplitudes:
+ + . . .
quantization via “Feynman-diagrams”
scattering amplitude: A =∞∑
n=0
A(n)g2+2ns .
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Results:
➪ spectrum contains non-Abelian gauge theory
with families of chiral fermions coupled to gravity
➪ for scattering processes with p � MS :
string theoryp�MS
−→ QFT & GR (Astring −→ AQFT,GR)
⇒ QFT & GR are low energy limit of string theory.
➪ gs is free parameter and one can choose gs � 1
⇒ perturbative evaluation of A possible
➪ amplitudes A(n) are UV-finite ?
⇒ pert. quantum corrections to GR can be sensibly computed
➪string theory provides perturbative quantum gravity
and unifies all interactions
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Problems of perturbative String Theory
➪ spectrum is supersymmetric ⇒ necessary for consistency?
➪ quarks, leptons and Higgs massless
⇒ what generates small masses ?
⇒ what generates the hierarchy MZ
MPl
≈ 10−17
➪ unitarity of scattering amplitudes ⇒ D ≤ 10
• D = 10: 5 different string theories:
IIA, IIB, I, Het. E8 × E8, Het. SO(32)
• D < 10: families of theories
geometrical compactification: M(10) = M(D) × K(10−D)
⇒ what chooses D ? ⇒ what chooses K ?
hope: cured by non-perturbative effects.
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Non-perturbative Aspects of String Theory
conjecture:
different string theories are dual description of one quantum theory:
A B
gA << 1 gB
<< 1
perturbative spectrum A ⇔ non-perturbative spectrum B
perturbative spectrum B ⇔ non-perturbative spectrum A
difficult to prove but successful checks on ‘protected’ couplings
(such couplings do exist in supersymmetric theories)
Non-perturbative states of string theory: D-branes
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D-Branes[Polchinski]
open string with Dirichlet boundary condition define hyperplane
➪ D-Branes are dynamical objects of string theory
➪ D-Branes are non-perturbative states of string theory
(tension ∼ g−1s )
➪ string theory is not a theory of only strings but also
describes higher-dimensional objects: Branes
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M-Theory
Proposal: all string theories are pert. limits of one quantum theory
What is M-theory ?
Suggestion: theory of D-particles [Banks, Fischler, Shenker, Susskind].
⇒ space-time becomes non-commutative
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The Standard Model on a D3-brane
⇒ gauge interaction is localized on the D3-branes
⇒ transverse dimension only feel gravitational interaction
⇒ how big can they be?
V (r) ∼1
r
Coulomb valid down to ∼ 10−16cm
Newton valid down to ∼ 10−1cm
⇒ large extra dimension possible
⇒ signal at LHC ?
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Cosmology in String Theory
so far only little known about cosmological solutions in ST
⇒ no convincing picture of the Big Bang
➪ early universe: inflationary phase
ST: has many scalar fields ! with right couplings ?
➪ late universe: Λ > 0 ST: problematic asymptotically
options:
• time-dependent scalar field can ‘simulate’
Tµν = −Λgµν (quintessence)
• only local de-Sitter vacuum (string landscape)
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Black Holes in String Theory
With help of duality one can study black holes
• so far only supersymmetric black holes studied
(extreme Reissner-Nordstrøm BH: M = Q)
• Bekenstein–Hawking entropy
S =A
4GN
, A = Area of Horizon
can be computed as sum over micro-states [Strominger, Vafa]
micro-states = excitations of D-branes
• Hawking radiation (temperature und decay rate) is reproduced
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String theory as a technical tool
➪ use string theory to organize QCD amplitudes
[Bern, Dixon, Kosower, Witten]
➪ use string theory as regulator ⇒ lessons for supersym. QFT
➪ learn about strongly coupled (supersym.) QFTs ⇒ (s)QCD
• AdS/CFT correspondence (N = 4)
• Seiberg/Witten (N = 2)
• Dijkgraaf/Vafa (N = 1)
and develop new tools
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String Theory and Mathematics
➪ point-like particle ≡ probe of continuous space-time geometry
⇒ relation with Riemannian geometry [Einstein,Hilbert]
➪ string as probe: sees coarser structure
⇒ development of quantum geometry [Kontsevich, Manin, . . . ]
surprising results:
• mirror symmetry in Calabi-Yau manifolds
• computation of number of holomorphic curves on Calabi-Yau
manifolds
• development of quantum cohomology
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Summary
➪ GR and QFT are the foundations of contemporary physics
➪ GR/cosmology successfully describes the universe
QFT/particle physics successfully describes small length scales
➪ supersymmetric theories have remarkable properties:
• consistency with electro-weak precision experiments
• unification of the gauge couplings
• candidates for dark matter
• light neutrinos
➪ string theory unifies all interactions
• provides perturbative quantum gravity
• qualitative agreement with generalizations of the SM
• non-perturbative definition of string theory not yet achieved
Jan Louis
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